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Least Operators to Express Number

Difficulty: Hard

Problem Description

Given a single positive integer x, we will write an expression of the form x (op1) x (op2) x (op3) x ... where each operator op1, op2, etc. is either addition, subtraction, multiplication, or division (+, -, *, or /). We would like to write an expression with the least number of operators such that the expression equals the given target. Return the least number of operators used.

Key Insights

  • The operations will follow the usual order of operations: multiplication and division happen before addition and subtraction.
  • Division results in rational numbers, which can complicate the calculations.
  • The problem can be approached using dynamic programming or a breadth-first search (BFS) to explore different combinations of operations efficiently.
  • The goal is to minimize the number of operations used to reach the target.

Space and Time Complexity

Time Complexity: O(log(target) * log(target)) - due to the exploration of various operations and their combinations. Space Complexity: O(log(target)) - for storing intermediate results and states.

Solution

The solution utilizes a breadth-first search (BFS) approach to explore all possible expressions formed by repeatedly applying the operators on the integer x. We keep track of the current value and the number of operations used to reach that value. The BFS ensures we explore all potential combinations of operations and find the minimum number of operations needed to reach the target.

  1. Start with the initial value x.
  2. Use a queue to store tuples of the current value and the number of operations used.
  3. For each element in the queue, apply all four operations (addition, subtraction, multiplication, division) to generate new values.
  4. Check if the new value matches the target. If it does, return the count of operations.
  5. If not, continue exploring until all possibilities are exhausted.

Code Solutions

from collections import deque

def leastOperators(x: int, target: int) -> int:
    if x == target:
        return 0
    
    queue = deque([(x, 0)])  # (current value, number of operations)
    visited = set([x])  # to avoid visiting the same value again

    while queue:
        current, ops = queue.popleft()
        
        # Define the possible next values
        next_values = [
            current + x,
            current - x,
            current * x,
            current / x if current % x == 0 else None
        ]
        
        for next_val in next_values:
            if next_val is not None:
                if next_val == target:
                    return ops + 1
                if next_val not in visited and 0 < next_val <= 2 * 10**8:
                    visited.add(next_val)
                    queue.append((next_val, ops + 1))

    return -1  # If target is not reachable
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